A Wadge hierarchy for second countable spaces

Abstract

We define a notion of reducibility for subsets of a second countable T0 topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the Hausdorff–Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T0 space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line \({\mathbb{R}}\) or the Scott Domain \({\mathcal{P}\omega}\) .